After some hair pulling and bad language I've got something to work. It's very crude, but gives a glimpse at what can be done. If you visit http://iphylo.org/~rpage/mytreebase/ and enter a taxon name, my code paddles off and queries TreeBASE to see if it has any phylogenies for that taxon. Gears grind, RSS feeds are crunched, a triple store is populated, NEXUS files are grabbed and Newick trees extracted, small creatures are needlessly harmed, and at last some phylogeny thumbnails are rendered in SVG (based on code I mentioned earlier), grouped by study. Functionality is limited (you can't click on the trees to make them bigger, for example), and the bibliographic information TreeBASE stores for studies is a bit ropey, but you get the idea.
What I'm looking for at this stage is a very simple interface that answers the question "show me the trees", which I think is the most basic question you can ask of TreeBASE (and one its own web interface makes unnecessarily hard). I've also gained some inspiration from the BioText search engine.
If you want to give it a try, here are some examples. These examples should be fairly responsive as the data is cached, but if you try searching for other taxa you may have a bit of a wait while my code talks to TreeBASE.